Independent exponential random variable

Question

Let $X=X_1+...X_n$ where $X_i's$ are independent exponential random variables of parameter 1.

(a) Show that the moment generating function of $X_1$ is $\frac{1}{1-t}$

(b) Compute the moment generating function of $X$, $M(t)=E(e^{tx})$.

(c) Show that for any $a>0$, $P(X>a)\leq e^{-at}M(t)$

(d) Give an upper bound of $P(X>200)$.

For this question, I got (a), (b) $E(e^{tx})=\frac{1}{(1-t)^n}$ and (d) $P(X>200)\leq\frac{n}{200}$ by using Markov inequality. But I am unable to show for part (c). I am suspecting it to use Markov inequality but I was unable to do so. Any hints?

Answer 1

Hint: Show that $$\mathbb{P}(X>a) = \mathbb{P}(e^{tX}>e^{ta}), \qquad t>0,$$ for any $a>0$, and apply the Markov inequality.